Plot True Course and Distance Between Two Charted Points

TL;DR — On a Mercator chart, the straight line connecting two points is a rhumb line representing a constant true course; distance is measured using the latitude scale adjacent to the line, not the longitude scale. Bowditch Ch. 4 §402

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What the Rule Says

The foundation for plotting true course and distance on a nautical chart rests on the properties of the Mercator projection. Bowditch Ch. 4 §402 states that the Mercator projection presents lines of latitude and longitude as straight lines crossing at right angles, with rhumb lines — lines of constant bearing — appearing as straight lines. This is the single most important geometric fact for chart plotting on the OUPV and Master 100 GT exam.

Two consequences follow directly from that projection property:

First: True course. When you draw a straight line between two charted points on a Mercator chart, that line is a rhumb line. The angle that line makes with any meridian (a vertical line of longitude on the chart) is the true course. You read that angle using the compass rose printed on the chart, or by transferring the line to the nearest compass rose with parallel rulers or a course plotter. The result is expressed as a three-digit true bearing from 000° through 359°. Bowditch Ch. 4 §402

Second: Distance. Because meridian length varies with latitude on a Mercator chart — the chart is stretched progressively toward the poles — you must measure distance using the latitude (vertical) scale at the same latitude as the line being measured, not the longitude (horizontal) scale. One minute of latitude equals one nautical mile. Bowditch Ch. 4 §402 This rule is non-negotiable and is the source of the most common plotting errors on the exam.

Dead reckoning builds directly on these two measurements. Bowditch Ch. 1 §102 defines dead reckoning (DR) as the determination of position by applying course steered and distance run from a known starting point. The true course and distance you plot between two points are the inputs to every DR calculation that follows.

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Why It Matters on the Exam

The Mercator Projection and the Rhumb Line

Exam questions on this topic test whether you understand why a straight line on a Mercator chart gives you a usable course. The answer is the rhumb-line property of the Mercator projection: a constant-bearing track plots as a straight line. Bowditch Ch. 4 §402 If the chart were a different projection — a gnomonic chart, for example — a straight line would represent a great circle, not a rhumb line, and the procedure would be entirely different. The exam assumes a standard Mercator nautical chart unless otherwise stated.

Reading True Course from the Compass Rose

The compass rose on a NOAA chart has an outer ring graduated in true degrees and an inner ring graduated in magnetic degrees (with the local variation noted). For exam purposes, you always work in true unless the question specifically asks for magnetic or compass course. When you transfer your plotted line to the compass rose using parallel rulers:

• Walk the rulers carefully across the chart without allowing them to slip.
• Pass through the center of the nearest compass rose.
• Read the true course on the outer ring in the direction of travel.
• The reciprocal (opposite direction) reads 180° away on the same ring.

Every factual claim about plotting accuracy traces back to the principle that plotting errors are a recognized source of navigational error. Bowditch Ch. 23 §2301 lists plotting error explicitly among the categories of navigational error — instrument error, observation error, plotting error, computation error, and the assumption that the vessel has held the planned course. Sloppy parallel-ruler work or misreading the compass rose ring are plotting errors that compound over time.

Measuring Distance Correctly

This is where candidates lose points. The procedure:

1. Set your dividers to span the line between the two points, or to a convenient sub-interval if the line is long. 2. Transfer the dividers to the latitude scale (the left or right border of the chart) at the same latitude as the segment being measured. 3. Count the minutes of latitude. Each minute = 1 nautical mile.

Bowditch Ch. 4 §402 is explicit: distances must be measured at the latitude of the line, since meridian length varies with latitude. For a line that spans several degrees of latitude, measure in segments, using the mid-latitude of each segment against the latitude scale. Never use the longitude scale (top or bottom border) for distance — those divisions represent degrees of longitude, whose length in miles shrinks as latitude increases.

Connecting Course and Distance to DR

Once you have true course and distance, you have everything needed to advance a DR position. Bowditch Ch. 1 §102 establishes that DR accuracy degrades over time and must be confirmed by external fixes whenever possible. The exam will often chain a plotting question into a DR question: given a departure point, a true course, a speed, and an elapsed time, find the DR position. The distance run equals speed multiplied by time (D = S × T), and that distance is laid off along the true course line from the last known position.

Speed and distance instrumentation on vessels required to carry ARPA must meet the accuracy standards in 33 CFR §164.40: speed errors must not exceed 5 percent of vessel speed or 0.5 knot, whichever is greater, and distance errors must not exceed 5 percent of distance run per hour or 0.5 nautical mile per hour, whichever is greater. 33 CFR §164.40 These tolerances matter when evaluating how much confidence to place in an instrument-derived distance run versus a plotted DR.

Current, Set, and Drift

A plotted true course is the course through the water. It is not necessarily the course made good over the ground when current is present. Bowditch Ch. 23 §2302 defines set as the direction toward which the current flows and drift as the speed of the current. The vector sum of the course steered through the water and the current vector gives the actual course over ground. Bowditch Ch. 24 §2402 states the vector triangle relationship: course steered plus drift vector equals course made good. Exam questions may give you a plotted course between two points and ask what course to steer to make good that track in the presence of a known current — that requires inverting the triangle. Bowditch Ch. 24 §2402

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Common Pitfalls

1. Using the longitude scale to measure distance. The longitude scale runs along the top and bottom borders of the chart. Its divisions are degrees and minutes of longitude, not latitude. Because a minute of longitude equals one nautical mile only at the equator, using it at any other latitude will give a distance that is too short. Always use the latitude scale on the side borders. Bowditch Ch. 4 §402

2. Reading the wrong ring of the compass rose. The outer ring is true; the inner ring is magnetic. Exam questions are written in true unless stated otherwise. Reading the inner ring and reporting a magnetic course as a true course is a common error. The magnetic variation printed inside the compass rose is a clue that the inner ring is magnetic — do not use it for true course answers.

3. Misidentifying the direction of travel. A line between two points has two directions. When you transfer to the compass rose, make sure you read the direction from your departure point toward your destination, not the reciprocal. The reciprocal will be exactly 180° different. Label your course line with an arrow in the direction of travel to avoid this.

4. Failing to account for current when the question asks for course to steer. Plotting the straight line between two points gives you the course made good (the track over ground you want to achieve). If the question introduces a current, the course to steer through the water will differ. Bowditch Ch. 23 §2302 and Bowditch Ch. 24 §2402 govern that vector correction. Do not report the track as the course to steer when current is present.

5. Ignoring DR degradation over time. A plotted course and distance give you a DR track, not a guaranteed position. Bowditch Ch. 1 §102 and Bowditch Ch. 23 §2301 both emphasize that cumulative errors must be checked against fixes. On the exam, if a question offers you a fix opportunity, use it — do not extend a DR indefinitely.

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Quick Check

Q1: Why does a straight line on a Mercator chart represent a constant true course?

Because the Mercator projection renders rhumb lines — lines of constant bearing — as straight lines. The projection presents latitude and longitude as straight lines crossing at right angles, so any straight line drawn on the chart crosses every meridian at the same angle, which is the definition of a rhumb line. Bowditch Ch. 4 §402

Q2: Which scale do you use to measure distance on a Mercator chart, and why?

The latitude scale on the side borders of the chart. One minute of latitude equals one nautical mile. The longitude scale cannot be used because meridian length varies with latitude — a minute of longitude equals one nautical mile only at the equator and decreases toward the poles. Bowditch Ch. 4 §402

Q3: You plot a true course of 045° T between two points. The current sets 270° T at 2 knots. Will your vessel make good 045° T if you steer 045° T? What concept applies?

No. Steering 045° T through the water while a current sets 270° T will result in a course made good that differs from 045° T. The vector sum of the course steered and the current vector determines the actual course over ground. To make good 045° T, you must solve the current vector triangle and steer a corrected course. Bowditch Ch. 23 §2302, Bowditch Ch. 24 §2402

Q4: What is the maximum allowable speed error for a speed/distance device on a vessel required to carry ARPA, when operating free from shallow water, wind, current, and tide?

The greater of 5 percent of the vessel's speed or 0.5 knot. 33 CFR §164.40

Q5: You depart a known position on a true course of 180° T at 10 knots. After 90 minutes with no fix, what is your DR position, and how confident should you be in it?

Distance run = 10 knots × 1.5 hours = 15 nautical miles. The DR position is 15 miles due south of the departure point along the 180° T course line. Confidence should be limited — DR accuracy degrades over time due to instrument error, plotting error, and the assumption that the vessel has held the planned course. The DR must be confirmed by an external fix whenever possible. Bowditch Ch. 1 §102, Bowditch Ch. 23 §2301